We had a very simlair question, so I'll quote Garth (even though I strongly disagree with his conclusion the CMBR could provide a frame that is 'prefered' by the laws of physics, as there is no reason to think this is so) as I know that the 390 km/sec for the solar system is approx. correct:

When the Earth's velocity around the Sun has been taken into account the Solar System is travelling at 390 +- 60 km/sec relative to the surface of last emission of the CMB. However when the Sun's motion around the Galaxy is also taken into account this translates into the fact that the Galaxy is travelling relative to the surface of last emission of the CMB, which probably defines the C.M. reference frame of the universe, at 603 km/sec or about 0.2%c! (Nature, Vol 270, 3 Nov 1977, pg 9)

how quickly is the milky way relative to the cosmic backround radiation on its way through the universe ?

I mean not the rotation speed

thank you

hello king, you should have asked the direction as well
I have a fairly recent (2002) online source that may help.
http://arxiv.org/astro-ph/0210165 [Broken]

This source is a journal article (Phys. Rev. Series D, 2003) reporting an earlier measurment (Smoot et al) of the speed and direction of the Local Group (as usual wrt CMB)
the result IIRC is that motion is in direction of the constellation Crater
(a small faint constell in southern half of sky) or if you want a brighter marker try Corvus. And it gives the speed as 627+/-22 km/sec.
Part of this could be due, instead, to temperature fluctuation in the CMB itself but not enough to affect the rough idea of about 600 km/s.

the motion of Milky Way wrt center of mass of Local Group is small. It would not be too bad an approximation just to say equate Milky's speed wrt CMB and that of the Local Group. So one can say Milky is going about 600 km/s and in the direction of constellation Crater.

(the constellation stars just mark the direction in space, the other stars in the galaxy are not getting closer to them because they are moving along with the galaxy as well)

I am posting from memory, so will have to check the online source I gave to make sure.

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I see in another thread Nereid gave a link to an Astronomy Picture of the Day which gives a speed for Local Group. Nereid's APOD does not give speed of Milky, but it says speed of Local Group is about 600 km/s. Again you can assume Milky speed is about the same.https://www.physicsforums.com/showthread.php?p=303392#post303392

Also here jcsd quotes a Garth post which cites a Nature article roughly to the same effect----about 600 km/s. However there is no link. Maybe the Nature article is not online.

the celestial coords given for the Local Group velocity vector are
276 degrees, -33 degrees
so it is in the south hemisphere but not as far south as people usually associate with hydra and centaurus. the "Great Attractor" is in hydra/centaurus. therefore Local Group is not plunging directly at the Great Attractor. Virgo cluster might be, I dont know for sure, but Local deviates a bit from that.